Abstract To reduce the pollution from the atmosphere or polluted cities like the capital city Delhi of India, use of artificial rain is a solution. In this paper, we have proposed and analyzed a nonlinear mathematical model to reduce the pollution level by rain making. In the proposed model five variables are considered, namely; (i) number density of water vapor, (ii) number density of cloud drops

Abstract We consider a nonlinear filtering problem of multiscale non-Gaussian signal processes and observation processes with jumps. First, we prove that the dimension for the signal system can be reduced by a homogenized approach. Second, convergence of the corresponding nonlinear filtering to the homogenized filtering is shown by a weak convergence technique. Finally, we give an example to explain

Abstract In this work, we consider two dimensional stochastic tidal dynamics equations in bounded domains. We establish the exponential estimates on certain exit times associated with the solution trajectory of the tidal dynamics system. Using contraction principle, we study these exponential estimates of exit times in the context of a Wentzell-Freidlin type large deviations principle.

Abstract In this study, fractional stochastic pseudo-parabolic equations driven by fractional Brownian motion are investigated. This work aims at establishing existence, uniqueness, regularity results for mild solutions to an initial value problem for considered equations in two cases of H that are H>12 and H<12. In addition, the continuities of mild solutions with respect to the time variable and

Abstract In this article, based on the existing literature, we extend a deterministic delayed SIS epidemic model with vaccination to the stochastic version. First, we show that the solution of system is globally unique, positive and stochastically bounded. Then, constructing suitable Lyapunov function, we study the stochastic stability of system at the disease-free equilibrium. Moreover, the threshold

Abstract In this paper, we introduce and study a time-changed version of the space-time fractional Poisson process (STFPP) by time changing it by an independent Lévy subordinator with finite moments of any order. We call the introduced process as the time-changed fractional Poisson process (TCFPP). The probability mass function, probability generating function and a variant of the law of iterated logarithm

Abstract We discuss estimation of parameters for models governed by a stochastic differential equation driven by a mixed fractional Brownian motion with Gaussian random effects based on discrete observations.

Abstract In this paper, we propose and study a higher order stochastically perturbed SICA epidemic model for HIV transmission. We obtain sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of positive solutions to the system by establishing a series of suitable Lyapunov functions. In a biological point of view, the existence of a stationary distribution indicates

Abstract Novel coronavirus has altered the socio-economic condition of the whole world through its devastating effects on the human population. Mathematical models and computation techniques may play an important role in understanding this epidemic and contribute a lot in policy making to control the infection in a more systematic and effective way. In this paper, we have proposed a deterministic mathematical

Abstract In this work, we study the controllability of a class of stochastic integrodifferential equations with noncompact semigroups and nonlocal condition on Hilbert spaces.We suppose that the linear part admits a resolvent operator in the sense of Grimmer. Our main results are obtained by utilizing stochastic analysis theory, Kuratowski measure of noncompactness and Mönch fixed point Theorem. As

Abstract In this article, we establish the LaSalle-type theorem for stochastic functional differential equations with Markovian switching (SFDEwMSs) under much weaken conditions. We would like to emphasize that we do not require the linear growth condition and the bounded moment condition on the solutions. Indeed, we allow the Lyapunov function operator could be dependent of time to cover a much wider

Abstract This paper addresses the global output feedback regulation problem for a class of uncertain feedforward stochastic nonlinear systems. Unlike the previous works, the drift’s growth rate depends not only on an unknown constant but also on an output polynomial function and an input function. Moreover, the diffusion’s growth rate depends on an unknown constant or an output polynomial function

Abstract In a recent paper by Yu (arXiv:2008.05633, 2020), higher order derivatives of self-intersection local time of fractional Brownian motion were defined, and existence over certain regions of the Hurst parameter H was proved. Utilizing the Wiener chaos expansion, we provide new proofs of Yu’s results and show how a Varadhan-type renormalization can be used to extend the range of convergence for

Abstract In this paper, we study the two-person zero-sum stochastic games for controlled continuous time Markov chains with risk-sensitive finite-horizon cost criterion. The transition and cost rates are possibly unbounded. For the zero-sum risk-sensitive stochastic game, we prove the existence of the value of the game and a Markov saddle-point equilibrium in the class of all history-dependent multi-strategies

Abstract In this paper we study the discrete approximation of backward stochastic differential equations. Under a kind of non-Lipschitz assumption we prove the convergence of the proposed discrete scheme to the solution of backward stochastic differential equation.

Abstract We obtain the distribution of the sum of independent and non-identically distributed generalized Mittag–Leffler random variables. We then apply this result to study some related fractional point processes. We present their explicit probability mass functions as well as their connections with the fractional integral/differential equations. In the case of a point process with Mittag–Leffler

Abstract Let {X(t),t≥0} be a Wiener process with infinitesimal mean μ and variance 1, starting at X(0)∈[0,1]. Assume that X(0) is a random variable and define τ to be the first time that X(t) = 0 or 1. We look for probability density functions of X(0) such that P[X(τ)=0]=q∈(0,1). Various cases are considered both when μ = 0 and μ≠0.

Abstract Viruses are responsible of illness. The purpose of drug therapy is to fight bacterial infection and to achieve definite outcomes that improve the patient’s quality of life. Mathematical models combined with clinical studies can be useful in forecasting infection and understanding viral infections mechanism among host cells. In this work, incorporating stochastic fluctuations, we consider a

Abstract We prove an existence and uniqueness result of mild solution for a system of stochastic semilinear differential equations with fractional Brownian motions and Hurst parameter H < 1∕2. Our approach is based on Perov’s fixed point theorem, and we establish the transportation inequalities, with respect to the uniform distance, for the law of the mild solution.

Abstract We study zero-sum optimal stopping games associated with perpetual convertible bonds in an extension of the Black-Merton-Scholes model with random dividends under various information flows. In this type of contracts, the writers have the right to withdraw the bonds, before the holders convert them into assets. We derive closed-form expressions for the associated value function and optimal

Abstract In this contribution, a stochastic nonlinear evolution system under Neumann boundary conditions is investigated. Precisely, we are interested in finding an existence and uniqueness result for a random heat equation coupled with a Barenblatt’s type equation with a multiplicative stochastic force in the sense of Itô. In a first step, we establish well-posedness in the case of an additive noise

Abstract Approximate Bayesian computation (ABC) is a popular technique for approximating likelihoods and is often used in parameter estimation when the likelihood functions are analytically intractable. In the context of Hidden Markov Models (HMMs), we analyze the asymptotic behavior of the posterior distribution in ABC based Bayesian parameter estimation. In particular we show that Bernstein-von Mises

Abstract We present a new probabilistic analysis of distributed systems. Our approach relies on the theory of quasi-stationary distributions (QSD) and the results recently developed by the first and third authors. We give properties on the deadlock time and the distribution of the model before deadlock, both for discrete and diffusion models. Our results apply to any finite values of the involved parameters

Abstract A theoretical approach for solving time-fractional stochastic Ginzburg–Landau equation with mixed fractional Brownian motion in Hilbert space is elaborated. Initially, the stochastic partial differential system is reformulated in the Hilbert space by using the properties of fractional order space and fractional Laplacian. We establish the existence of mild solutions by employing Mittag–Leffler

Abstract In this paper, a Haar wavelet method is proposed for solving linear and nonlinear stochastic Itô–Volterra integral equation (SIVIE) driven by fractional Brownian motion (FBM) with Hurst parameter H ∈ ( 1 2 , 1 ) . This approach reduces the solution of the problem under study to the solution of a linear and nonlinear system of algebraic equations, which is based on the operational matrix and

Abstract In this article, we investigate a zero-sum stochastic game for continuous-time Markov chain with denumerable state space and unbounded transition rates, under the probability criterion. Under suitable assumptions, we show the existence of value of the game and also characterize it as the unique solution of a pair of Shapley equations. We also establish the existence of a randomized stationary

Abstract We generalize the Kolmogorov continuity theorem and prove the continuity of a class of stochastic fields with the parameter. As an application, we derive the continuity of solutions for nonlocal stochastic parabolic equations driven by non-Gaussian Lévy noises.

Abstract We consider an investor whose portfolio consists of a single risky asset and a risk free asset. The risky asset’s return has a heavy tailed distribution and thus does not have higher order moments. Hence, she aims to maximize the expected utility of the portfolio defined in terms of the median return. This is done subject to managing the Value at Risk (VaR) defined in terms of a high order

Abstract The mean time to finish a project is studied where the rate of work varies randomly. It is proved that the mean time to finish is bounded below by the time taken to complete the project if the work is performed at a fixed average rate, specifically, at the average work rate of the corresponding unrestricted or non-exit problem. Several examples illustrate the usefulness of the result in a

Abstract The main goal of this work is to relate weak and pathwise mild solutions for parabolic quasilinear stochastic partial differential equations (SPDEs). Extending in a suitable way techniques from the theory of nonautonomous semilinear SPDEs to the quasilinear case, we prove the equivalence of these two solution concepts.

Abstract Contributions of the present paper consist of two parts. In the first one, we contribute to the theory of stochastic calculus for signed measures. For instance, we provide some results permitting to characterize martingales and Brownian motion both defined under a signed measure. We also prove that the uniformly integrable martingales (defined with respect to a signed measure) can be expressed

Abstract We provide a probabilistic SIRD model for the COVID-19 pandemic in Italy, where we allow the infection, recovery and death rates to be random. In particular, the underlying random factor is driven by a fractional Brownian motion. Our model is simple and needs only some few parameters to be calibrated.

Abstract The success in reducing global HIV prevalence rates is attributed to control measures like information and education campaigns (IECs), antiretroviral therapy (ART), and national, multinational and multilateral support providing official developmental assistance (ODAs) to combat HIV. A class of stochastic nonlinear multi-population behavioral HIV/AIDS models is studied, where behavioral change

Abstract We give a particle picture interpretation of two recently discovered classes of self-similar stable processes with stationary increments studied by Samorodnitsky et al. and Dombry and Guillotin-Plantard. We study the occupation times of certain Poissonian systems of particles with ±1 charges and i.i.d. heavy-tailed weights, moving independently according to Lévy processes. We also obtain a

Abstract In this article, we study risk-sensitive stochastic differential games for controlled reflecting diffusion processes in a smooth bounded domain. We analyze the ergodic cost evaluation criterion for both nonzero-sum games and zero-sum games. Using principal eigenvalue approach, we establish the existence of Nash/saddle-point equilibria for relevant cases.

Abstract We study the strong convergence of the Carathéodory numerical scheme for a class of nonlinear McKean-Vlasov stochastic differential equations (MVSDE). We prove, under Lipschitz assumptions, the convergence of the approximate solutions to the unique solution of the MVSDE. Moreover, we show that the result remains valid, under continuous coefficients, provided that pathwise uniqueness holds

Abstract The first-passage time τ S ( x ) of a one-dimensional continuous stochastic process X ( t ) , starting from x ≤ S ( 0 ) , through a smooth boundary S(t) is investigated; in particular, diffusions and some kinds of Gaussian processes, such as Gauss-Markov and their fractional integrals, are considered. The tail behavior of P ( max s ∈ [ 0 , t ] X ( s ) > R ) and related asymptotics for τ S

Abstract Within the rough path framework, we prove the continuity of the solution to random differential equations driven by a one dimensional fractional Brownian motion with respect to the Hurst parameter H when H ∈ ( 1 / 3 , 1 / 2 ] .

Abstract The global well-posedness as well as long-term behavior in terms of mean random attractors and invariant measures are investigated for a class of stochastic discrete reaction-diffusion equations defined on Z k with a family of superlinear noise. The existence and uniqueness of weak pullback mean random attractors for the mean random dynamical system associated with the non-autonomous equations

Abstract The article is devoted to the existence and Hyers-Ulam stability of the almost periodic solution to the fractional differential equation with impulse and fractional Brownian motion under nonlocal condition. The investigation is mainly based on the semigroups of operators method and Mönch fixed points method, as well as the basic theory of Hyers-Ulam stability.

Abstract We study the asymptotic behavior of a real-valued diffusion whose nonregular drift is given as a sum of a dissipative term and a bounded measurable one. We prove that two trajectories of that diffusion converge almost sure to one another at an exponential explicit rate as soon as the dissipative coefficient is large enough. A similar result in Lp is obtained.

In this paper, we discuss a class of stochastic nonlinear partial differential equation of Burgers type driven by pseudo differential operator ( Δ + Δ α ) for α ∈ ( 0 , 2 ) , and fractional Brownian sheet. The existence and uniqueness of an Lp-valued (local) solution is established for the initial valued problem to the equation.

We investigate the joint distribution and the multivariate survival functions for the maxima of an Ornstein-Uhlenbeck (OU) process in consecutive time-intervals. A PDE method, alongside an eigenfunction expansion is adopted, with which we first calculate the distribution and the survival functions for the maximum of a homogeneous OU-process in a single interval. By a deterministic time-change and a

Abstract The aim of this manuscript is to analyze the existence of mild solution of non-instantaneous impulsive Hilfer fractional stochastic differential equations (NIHFSDEs) driven by fractional Brownian motion (fBm). Sufficient conditions for a class of NIHFSDEs of order 0<β<1 and of type 0≤α≤1 driven by fBm is derived with the help of fractional calculus, stochastic theory, fixed point theorem and

Abstract In the paper, we consider some piecewise deterministic Markov process, whose continuous component evolves according to semiflows, which are switched at the jump times of a Poisson process. The associated Markov chain describes the states of this process directly after the jumps. Certain ergodic properties of these two dynamical systems have been already investigated in our recent papers. We

Abstract We introduce and study a marked Poisson random measure on a σ-compact Hausdorff space. The underlying parameters of this measure are changing in accordance with the evolution of some stochastic process. This random measure (also known as a modulated measure) resembles those of the conventional Poisson random measure. Among many applications, the one about Poisson random measures modulated

Abstract Let W denote the Brownian motion. For any exponentially bounded Borel function g the function u defined by u(t,x)=E[g(x+σWT−t)] is the stochastic solution of the backward heat equation with terminal condition g. Let un(t,x) denote the corresponding approximation generated by a simple symmetric random walk with time steps 2T/n and space steps ±σT/n where σ>0. For a class of terminal functions

Abstract Kolmogorov’s converse inequality is proved for dependent random variables in order to obtain necessary and sufficient conditions for weak and strong laws (for not necessarily strictly stationary sequences) under strong dependence without rate assumptions.

Abstract Large deviation principle by the weak convergence approach is established for the stochastic nonlinear Schrödinger equation in one-dimension and as an application the exit problem is investigated.

Abstract We consider a new class of one-dimensional diffusion processes with self-similar random potentials. The self-similar random potential has different exponents to the left and the right hand sides of the origin. We show that, because of the difference between the two exponents, the long-time behaviors of our process on the left and the right hand sides of the origin are quite different from

Abstract On the global platform of emerging infectious diseases, dengue fever added a serious health concern, especially in the tropical and subtropical countries with poor health services. Due to unavailability of proper vaccination, primary prevention of dengue is possible at social and personal levels by controlling the propagation of vectors as well as avoiding the bites of infected ones, respectively

Abstract We formulate ill-posedness of inverse problems of estimation and prediction of Coronavirus Disease 2019 (COVID-19) outbreaks from statistical and mathematical perspectives. This is by nature a stochastic problem, since e.g., random perturbation in parameters can cause instability of estimation and prediction. This leaves us with a plenty of possible statistical regularizations, thus generating

Abstract This paper is concerned with existence, uniqueness, and approximate controllability of mild solutions to second-order non-autonomous stochastic impulsive differential systems. The impulsive differential systems may provide mathematical models to the phenomena having discontinuous jumps. The main results are carried out by using the generalized Banach contraction principle, evolution family

Abstract We study optimal control for mean-field forward–backward stochastic differential equations with payoff functionals of mean-field type. Sufficient and necessary optimality conditions in terms of a stochastic maximum principle are derived. As an illustration, we solve an optimal portfolio with mean-field risk minimization problem.

We prove an anticipative sufficient stochastic minimum principle in a jump process setup with initially enlarged filtrations. We apply the result to several portfolio selection problems like mean and minimal variance hedging under enlarged filtrations. We also investigate utility maximizing portfolio selection under future information. Contrarily to classical optimization methods like dynamic programing

In this article, we derive the risk-neutral measures of the stock options price and volatility by incorporating a simple constrained minimization of the Kullback measure of relative information. We obtain a second-order parabolic partial differential equation, the generalized Black–Scholes equation based on the theoretical analysis when the underlying financial asset is estimated using a stochastic

In this paper, stochastic integral equations characterized by fractional Brownian motion have been studied. The fractional stochastic integral equation has been solved by second kind Chebyshev wavelets. The convergence and error analysis have been discussed for the efficiency of the discussed method. In addition, two illustrative examples have been solved to examine the efficiency and accuracy of the

Abstract In this paper, stochastic integral equations characterized by fractional Brownian motion have been studied. The fractional stochastic integral equation has been solved by second kind Chebyshev wavelets. The convergence and error analysis have been discussed for the efficiency of the discussed method. In addition, two illustrative examples have been solved to examine the efficiency and accuracy

Abstract We prove an anticipative sufficient stochastic minimum principle in a jump process setup with initially enlarged filtrations. We apply the result to several portfolio selection problems like mean and minimal variance hedging under enlarged filtrations. We also investigate utility maximizing portfolio selection under future information. Contrarily to classical optimization methods like dynamic

Abstract In this article, we derive the risk-neutral measures of the stock options price and volatility by incorporating a simple constrained minimization of the Kullback measure of relative information. We obtain a second-order parabolic partial differential equation, the generalized Black–Scholes equation based on the theoretical analysis when the underlying financial asset is estimated using a stochastic